YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ) Problem 1: Dependency Pairs Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 0#(0(x1:S)) 0#(0(5(1(5(x1:S))))) -> 0#(x1:S) 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(5(x1:S)))) -> 0#(4(1(0(5(x1:S))))) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(1(2(0(x1:S)))) 0#(0(1(x1:S))) -> 0#(1(2(4(2(0(x1:S)))))) 0#(0(1(x1:S))) -> 0#(2(0(x1:S))) 0#(0(1(x1:S))) -> 0#(3(1(0(x1:S)))) 0#(0(1(x1:S))) -> 0#(4(0(x1:S))) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(0(2(1(x1:S)))) -> 0#(2(0(1(x1:S)))) 0#(5(1(x1:S))) -> 0#(2(3(1(5(x1:S))))) 0#(5(1(x1:S))) -> 0#(3(1(5(x1:S)))) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 0#(3(5(4(5(x1:S))))) 0#(5(3(4(1(x1:S))))) -> 5#(4(5(x1:S))) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 0#(1(3(0(4(5(x1:S)))))) 0#(5(4(0(1(x1:S))))) -> 0#(4(5(x1:S))) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(0(1(x1:S)))) -> 0#(2(0(1(x1:S)))) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 5#(1(2(0(x1:S)))) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 5#(1(2(4(0(x1:S))))) 5#(3(0(1(x1:S)))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 5#(1(2(3(0(x1:S))))) 5#(3(1(5(x1:S)))) -> 5#(3(1(2(5(x1:S))))) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) Problem 1: SCC Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 0#(0(x1:S)) 0#(0(5(1(5(x1:S))))) -> 0#(x1:S) 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(5(x1:S)))) -> 0#(4(1(0(5(x1:S))))) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(1(2(0(x1:S)))) 0#(0(1(x1:S))) -> 0#(1(2(4(2(0(x1:S)))))) 0#(0(1(x1:S))) -> 0#(2(0(x1:S))) 0#(0(1(x1:S))) -> 0#(3(1(0(x1:S)))) 0#(0(1(x1:S))) -> 0#(4(0(x1:S))) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(0(2(1(x1:S)))) -> 0#(2(0(1(x1:S)))) 0#(5(1(x1:S))) -> 0#(2(3(1(5(x1:S))))) 0#(5(1(x1:S))) -> 0#(3(1(5(x1:S)))) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 0#(3(5(4(5(x1:S))))) 0#(5(3(4(1(x1:S))))) -> 5#(4(5(x1:S))) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 0#(1(3(0(4(5(x1:S)))))) 0#(5(4(0(1(x1:S))))) -> 0#(4(5(x1:S))) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(0(1(x1:S)))) -> 0#(2(0(1(x1:S)))) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 5#(1(2(0(x1:S)))) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 5#(1(2(4(0(x1:S))))) 5#(3(0(1(x1:S)))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 5#(1(2(3(0(x1:S))))) 5#(3(1(5(x1:S)))) -> 5#(3(1(2(5(x1:S))))) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(5(1(5(x1:S))))) -> 0#(0(x1:S)) 0#(0(5(1(5(x1:S))))) -> 0#(x1:S) 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) ->->-> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 0#(0(x1:S)) 0#(0(5(1(5(x1:S))))) -> 0#(x1:S) 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) -> Usable rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [5](X) = X [1](X) = 2.X + 2 [2](X) = X [3](X) = X [4](X) = X [0#](X) = 2.X + 1 [5#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 0#(x1:S) 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(5(1(5(x1:S))))) -> 0#(x1:S) 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) ->->-> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 0#(x1:S) 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) -> Usable rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [5](X) = X + 1 [1](X) = X + 2 [2](X) = X [3](X) = X [4](X) = X [0#](X) = 2.X + 2 [5#](X) = X + 1 Problem 1: SCC Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) ->->-> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 5#(0(0(x1:S))) 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) -> Usable rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [5](X) = X + 1 [1](X) = X + 2 [2](X) = X [3](X) = X [4](X) = X [0#](X) = 2.X + 1 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) ->->-> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(0(5(1(5(x1:S))))) -> 5#(5(0(0(x1:S)))) 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) -> Usable rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [5](X) = X [1](X) = 2.X + 2 [2](X) = X [3](X) = X [4](X) = X [0#](X) = 2.X + 2 [5#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) ->->-> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(0(1(5(x1:S)))) -> 0#(5(x1:S)) 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) -> Usable rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [5](X) = X [1](X) = 2.X + 2 [2](X) = X [3](X) = X [4](X) = X [0#](X) = 2.X [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) ->->-> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(x1:S)) 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) -> Usable rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [5](X) = X + 1 [1](X) = X + 2 [2](X) = X [3](X) = X [4](X) = X [0#](X) = 2.X + 1 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) ->->-> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) Problem 1: Subterm Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(x1:S) 0#(0(2(1(x1:S)))) -> 0#(1(x1:S)) 0#(5(1(x1:S))) -> 5#(x1:S) 0#(5(3(4(1(x1:S))))) -> 5#(x1:S) 0#(5(4(0(1(x1:S))))) -> 5#(x1:S) 0#(1(1(1(x1:S)))) -> 0#(x1:S) 0#(1(1(x1:S))) -> 0#(x1:S) 5#(0(1(1(x1:S)))) -> 0#(x1:S) 5#(0(1(x1:S))) -> 0#(x1:S) 5#(3(0(1(x1:S)))) -> 0#(x1:S) -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Projection: pi(0#) = 1 pi(5#) = 1 Problem 1: SCC Processor: -> Pairs: Empty -> Rules: 0(0(5(1(5(x1:S))))) -> 1(2(5(5(0(0(x1:S)))))) 0(0(1(5(x1:S)))) -> 0(4(1(0(5(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(1(2(x1:S)))))) 0(0(1(x1:S))) -> 0(1(2(0(x1:S)))) 0(0(1(x1:S))) -> 0(1(2(4(2(0(x1:S)))))) 0(0(1(x1:S))) -> 0(1(3(0(2(x1:S))))) 0(0(1(x1:S))) -> 0(1(3(0(4(x1:S))))) 0(0(1(x1:S))) -> 0(2(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(1(2(x1:S))))) 0(0(1(x1:S))) -> 0(3(0(3(1(x1:S))))) 0(0(1(x1:S))) -> 0(3(1(0(x1:S)))) 0(0(1(x1:S))) -> 0(4(0(4(1(x1:S))))) 0(0(1(x1:S))) -> 1(0(4(0(x1:S)))) 0(0(1(x1:S))) -> 1(2(0(2(0(x1:S))))) 0(0(1(x1:S))) -> 1(2(0(3(0(4(x1:S)))))) 0(0(1(x1:S))) -> 1(2(2(0(0(x1:S))))) 0(0(2(1(x1:S)))) -> 2(0(3(0(2(1(x1:S)))))) 0(0(2(1(x1:S)))) -> 2(3(0(2(0(1(x1:S)))))) 0(5(1(x1:S))) -> 0(2(3(1(5(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(2(x1:S))))) 0(5(1(x1:S))) -> 0(3(1(5(x1:S)))) 0(5(1(x1:S))) -> 0(3(1(2(5(2(x1:S)))))) 0(5(1(x1:S))) -> 0(4(5(1(x1:S)))) 0(5(3(0(1(x1:S))))) -> 1(0(5(3(0(4(x1:S)))))) 0(5(3(4(1(x1:S))))) -> 1(0(3(5(4(5(x1:S)))))) 0(5(4(0(1(x1:S))))) -> 0(1(3(0(4(5(x1:S)))))) 0(1(0(1(x1:S)))) -> 1(0(2(0(1(x1:S))))) 0(1(1(1(x1:S)))) -> 1(1(3(1(0(x1:S))))) 0(1(1(x1:S))) -> 0(2(1(1(x1:S)))) 0(1(1(x1:S))) -> 0(3(1(1(x1:S)))) 0(1(1(x1:S))) -> 1(0(3(1(2(4(x1:S)))))) 0(1(1(x1:S))) -> 1(0(4(2(1(2(x1:S)))))) 0(1(1(x1:S))) -> 1(1(2(4(3(0(x1:S)))))) 0(1(1(x1:S))) -> 1(1(3(0(4(x1:S))))) 0(1(1(x1:S))) -> 1(2(0(2(1(x1:S))))) 0(1(1(x1:S))) -> 1(2(1(0(4(4(x1:S)))))) 0(1(1(x1:S))) -> 1(2(2(1(3(0(x1:S)))))) 5(0(1(1(x1:S)))) -> 1(5(1(2(0(x1:S))))) 5(0(1(x1:S))) -> 5(0(2(1(2(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(3(0(4(x1:S)))))) 5(0(1(x1:S))) -> 5(1(2(4(0(x1:S))))) 5(3(0(1(x1:S)))) -> 5(1(2(3(0(x1:S))))) 5(3(1(5(x1:S)))) -> 5(3(1(2(5(x1:S))))) 5(3(2(1(x1:S)))) -> 1(2(3(5(2(x1:S))))) 5(4(0(1(x1:S)))) -> 1(2(5(0(4(x1:S))))) 5(4(2(1(1(x1:S))))) -> 5(4(1(2(1(2(x1:S)))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.